Wallpaper Groups

For this post, I am going to be completing the ‘doing math’ requirement.

I completed the 17 wallpaper groups as part of my project for this course and decided to complete a blog post on the work that I completed! The 17 wallpaper groups represent the seventeen different ways to cover a two-dimensional plane if one only uses symmetries. Each individual group is a collection of these symmetries that are used in unique ways, making each group different from one another. First, I wanted to define the different types of symmetries used to tile the plane.
Rotation – a transformation in which an object is rotated about a specific point, typically rotated in degrees.
Reflection – a transformation in which a mirror image is created around an axis of reflection.
Translation – a transformation that moved every point in an object the same amount of distance.
Glide Reflection – a transformation that is a combination of a reflection and a translation.

All of these symmetries are important for understanding how these wallpaper groups have been created and how they move around the plane. Another important definition that is used to describe the movement in the plane is a lattice. This term describes all the centers of rotation and the axes of reflection for the objects in the plane. Now, we can continue on to show each of the seventeen wallpaper groups that my group and I have created for our project.

Symmetry Group 1 (p1):
This group is the simplest of all the symmetry groups, it consists only of translations throughout the plane. It does not contain reflections, rotations, or glide reflections and its lattice is parallelogrammatic.
Screen Shot 2015-12-14 at 12.59.25 PM

Symmetry Group 2 (p2):
Screen Shot 2015-12-14 at 1.00.38 PMThis group consists of translations and 180 degree rotations within the plane. It does not contain reflections or glide reflections. Also, the two translation axes may be inclined at any angle to each other. The lattice for this symmetry group is parallelogrammatic.

Symmetry Group 3 (pm):
Screen Shot 2015-12-14 at 1.02.33 PMThis group consists of reflections and translations. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. There are no rotations or glide reflections and the lattice is rectangular.

Symmetry Group 4 (pg):
Screen Shot 2015-12-14 at 1.04.18 PMThis symmetry group contains glide reflections and translations. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. There are no rotations or regular reflections and the lattice is rectangular.

Symmetry Group 5 (cm):
This group contains reflections, glide reflections, and translations.
Screen Shot 2015-12-14 at 1.06.49 PM
The reflections and glide reflections have parallel axes. There are no rotations in this group and the lattice is rhombic. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes. This group applies for symmetrically staggered rows of identical objects, which have a symmetry axis perpendicular to the rows.

Symmetry Group 6 (pmm):
Screen Shot 2015-12-14 at 1.09.24 PMThis group contains reflections and rotations whose axes are perpendicular. There are no glide reflections and the only rotations are half-turns whose fixed points lie at intersections of axes of reflection. The lattice for this symmetry group is rectangular.

Symmetry Group 7 (pmg):
Screen Shot 2015-12-14 at 1.11.16 PMThis symmetry group contains reflections and 180 degree rotations. It does not contain translations or glide reflections. The fixed points of the half turns do not lie on the axes of reflection. The lattice for this group is rectangular.

Symmetry Group 8 (pgg):
Screen Shot 2015-12-14 at 1.12.59 PM
This group contains translations, glide reflections, and 180 degree rotations but it does not contain regular reflections. There are perpendicular axes for the glide reflections, and the fixed points of the 180 degree rotations do not lie on these axes. The lattice for this group is rectangular.

Symmetry Group 9 (cmm):
Screen Shot 2015-12-14 at 1.15.24 PM
This group contains reflections and glide reflections. This group also contains half turns (rotations). This group has perpendicular reflection axes and has half turns. The lattice for this specific group is rhombic.

Symmetry Group 10 (p4):
This group contains a 90 degree rotation, a 180 degree rotation, and translations. The 90 degree rotation is an order 4 rotation, and the half turn is a order 2 rotation. The centers of the half turns are midway between the centers of the order 4 rotations. There are not reflections or glide reflections and the lattice is square.
Screen Shot 2015-12-14 at 1.17.05 PM

Symmetry Group 11 (p4m):
Screen Shot 2015-12-14 at 1.20.11 PMThis group contains reflections, 180 degree rotations, and 90 degree rotations. The axes of reflection are inclined to each other by 45 degrees so that four axes of reflection pass through the centers of the order 4 rotations. All rotation centers lie on the reflection aces. The lattice is square for this symmetry group.

Symmetry Group 12 (p4g):
Screen Shot 2015-12-14 at 1.22.01 PMThis group contains translations, reflections, glide reflections, 90 degree rotations, and 180 degree rotations. Different from symmetry group 11, the axes of reflection are perpendicular and none of the centers of the 90 degree rotation lie on the reflection axes. The lattice for this group is square.

Symmetry Group 13 (p3):
This group contains a 120 degree rotation.
Screen Shot 2015-12-14 at 1.24.41 PMThis rotation is of order 3. The lattice for this group is hexagonal. There are no reflections or glide reflections. The rotation centers can be found at the corners of these triangles and at the centers of them.

Symmetry Group 14 (p31m):

Screen Shot 2015-12-14 at 1.26.33 PM
This group contains reflections and rotations of order 3 (120 degrees). Some of the centers of rotation lie on the reflection axes, and some do not. The lattice is hexagonal.

 

 

Symmetry Group 15 (p3m1):
Screen Shot 2015-12-14 at 1.28.36 PM
This group contains translations, 120 degree rotations, glide reflections, and regular reflections. The axes of reflection are inclined at 60 degree to one another, but all centers of rotation lie on reflection axes. The lattice is hexagonal. The glide reflection is given after the translation and reflection have been done.

Symmetry Group 16 (p6):
This group contains 60 degree rotations (order 6), 180 degree rotations, and 120 degree rotations. This group does not contain reflections or glide reflections. The lattice is hexagonal. A fundamental region for the symmetry group is one-sixth of an equilateral triangle of the lattice.
Screen Shot 2015-12-14 at 1.31.43 PM

Symmetry Group 17 (p6m):
Screen Shot 2015-12-14 at 1.32.43 PMThis is the most complicated group consists of translations, reflections, glide reflections, 60 degree rotations, 120 degree rotations, and 180 degree rotations. The axes of reflection meet at all the centers of rotation. The lattice is hexagonal. The glide reflection is given after the translation and reflection have been applied.

Shown with examples that my group and I created, these are the 17 wallpaper groups that exist in the two-dimensional plane.

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